Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Solve the given initial value problem $y'' + 2 y' + 17 y = 0;$ $y (0) = 1,$ $y' (0) = - 1$ .

The solution of the given initial value $y'' + 2 y' + 17 y = 0$ is $y (t) = e^{- t} (\cos 4 t)$ when $y (0) = 1$ and $y' (0) = - 1 .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $α \pm i β$ , then the general solution is given as:

$y (t) = c_{1} e^{α t} c o s β t + c_{2} e^{α t} s i n β t$ .

Step 2: Finding the roots of the auxiliary equation.

Given differential equation is $y'' + 2 y' + 17 y = 0$

Then the auxiliary equation $r^{2} + 2 r + 17 = 0$

Solve the auxiliary equation to obtain the roots.

$\begin{matrix} r = \frac{- 2 \pm \sqrt{2^{2} - 4 \times 1 \times 17}}{2 \times 1} \\ r = \frac{- 2 \pm \sqrt{4 - 68}}{} \\ r = \frac{- 2 \pm \sqrt{- 64}}{} \\ r = \frac{- 2 \pm 8 i}{} \\ r = - 1 \pm 4 i \end{matrix}$

Therefore, the general solution is:

$\begin{matrix} y (t) = e^{- 1 \times t} (c_{1} \cos (4 t) + c_{2} \sin (4 t)) \\ = e^{- t} (c_{1} \cos (4 t) + c_{2} \sin (4 t)) \end{matrix}$

Step 3: Finding the values of C1 and C2

Given initial conditions are $y (0) = 1$ and $y' (0) = - 1 .$

role="math" localid="1654841671923" $\begin{array}{l} y (0) = e^{- 0} (c_{1} \cos (4 \times 0) + c_{2} \sin (4 \times 0)) \\ c_{1} = 1 \end{array}$

And

$y' (t) = - e^{- t} (c_{1} \cos 4 t + c_{2} \sin 4 t) + e^{- t} (- 4 c_{1} sint + 4 c_{2} cost)$

Then,

$\begin{array}{l} y' (0) = - e^{- 0} (c_{1} \cos (4 \times 0) + c_{2} \sin (4 \times 0)) + e^{- 0} (- 4 c_{1} \sin (4 \times 0) + 4 c_{2} \cos (4 \times 0)) \\ - c_{1} + 4 c_{2} = - 1 \end{array}$

Substitute $c_{1}$ in the above equation

$\begin{matrix} - 1 + c_{2} = - 1 \\ c_{2} = 0 \end{matrix}$

Therefore, the solution is $y (t) = e^{- t} (\cos 4 t)$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Solve the given initial value problem $y'' + 2 y' + 17 y = 0;$ $y (0) = 1,$ $y' (0) = - 1$ .

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the roots of the auxiliary equation.

Step 3: Finding the values of C1 and C2

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Solve the given initial value problem y''+2y'+17y=0;y(0)=1,y'(0)=-1.

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the roots of the auxiliary equation.

Step 3: Finding the values of C1 and C2

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Solve the given initial value problem $y'' + 2 y' + 17 y = 0;$ $y (0) = 1,$ $y' (0) = - 1$ .